| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To compute the sample variance of the set \(\{1, 12, -17, -3, -11, -5\}\), we follow these steps: 1. **Calculate the mean (average) of the sample:** \[ \text{Mean} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} \] \[ \text{Mean} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} \] \[ \text{Mean} = \frac{-23}{6} \approx -3.8333 \] 2. **Calculate the squared differences from the mean for each data point:** \[ (1 - (-3.8333))^2 = (1 + 3.8333)^2 = 4.8333^2 \approx 23.36 \] \[ (12 - (-3.8333))^2 = (12 + 3.8333)^2 = 15.8333^2 \approx 250.69 \] \[ (-17 - (-3.8333))^2 = (-17 + 3.8333)^2 = -13.1667^2 \approx 173.36 \] \[ (-3 - (-3.8333))^2 = (-3 + 3.8333)^2 = 0.8333^2 \approx 0.69 \] \[ (-11 - (-3.8333))^2 = (-11 + 3.8333)^2 = -7.1667^2 \approx 51.36 \] \[ (-5 - (-3.8333))^2 = (-5 + 3.8333)^2 = -1.1667^2 \approx 1.36 \] 3. **Sum the squared differences:** \[ \sum (x_i - \bar{x})^2 = 23.36 + 250.69 + 173.36 + 0.69 + 51.36 + 1.36 \] \[ \sum (x_i - \bar{x})^2 = 500.82 \] 4. **Divide the sum of squared differences by \(n-1\) (where \(n\) is the number of data points):** \[ \text{Sample Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{500.82}{5} = 100.164 \] Thus, the sample variance is: \[ \boxed{100.164} \] |
\frac{601}{6} |
| deepseek-r1 | To compute the sample variance of the dataset \(\{1, 12, -17, -3, -11, -5\}\), we follow these steps: 1. **Calculate the sample mean**: \[ \text{Mean} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} = \frac{-23}{6} \] 2. **Compute the squared differences from the mean**: - For \(1\): \(\left(1 - \left(-\frac{23}{6}\right)\right)^2 = \left(\frac{29}{6}\right)^2 = \frac{841}{36}\) - For \(12\): \(\left(12 - \left(-\frac{23}{6}\right)\right)^2 = \left(\frac{95}{6}\right)^2 = \frac{9025}{36}\) - For \(-17\): \(\left(-17 - \left(-\frac{23}{6}\right)\right)^2 = \left(-\frac{79}{6}\right)^2 = \frac{6241}{36}\) - For \(-3\): \(\left(-3 - \left(-\frac{23}{6}\right)\right)^2 = \left(\frac{5}{6}\right)^2 = \frac{25}{36}\) - For \(-11\): \(\left(-11 - \left(-\frac{23}{6}\right)\right)^2 = \left(-\frac{43}{6}\right)^2 = \frac{1849}{36}\) - For \(-5\): \(\left(-5 - \left(-\frac{23}{6}\right)\right)^2 = \left(-\frac{7}{6}\right)^2 = \frac{49}{36}\) 3. **Sum the squared differences**: \[ \frac{841}{36} + \frac{9025}{36} + \frac{6241}{36} + \frac{25}{36} + \frac{1849}{36} + \frac{49}{36} = \frac{18030}{36} \] 4. **Divide by \(n-1\) (where \(n = 6\))**: \[ \text{Sample Variance} = \frac{18030 / 36}{5} = \frac{18030}{180} = \frac{601}{6} \] Thus, the sample variance of the dataset is \(\boxed{\dfrac{601}{6}}\). |
\frac{601}{6} |
| gpto1 | To find the sample variance of the given data set \{1, 12, -17, -3, -11, -5\}, we'll follow these steps: 1. **Calculate the sample mean (\(\overline{x}\))**: \[ \overline{x} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} = \frac{-23}{6} \] 2. **Find each squared deviation from the mean**: \[ \begin{align*} (1 - \overline{x})^2 &= \left(1 - \left(-\frac{23}{6}\right)\right)^2 = \left(\frac{29}{6}\right)^2 = \frac{841}{36} \\ (12 - \overline{x})^2 &= \left(12 - \left(-\frac{23}{6}\right)\right)^2 = \left(\frac{95}{6}\right)^2 = \frac{9025}{36} \\ (-17 - \overline{x})^2 &= \left(-17 - \left(-\frac{23}{6}\right)\right)^2 = \left(-\frac{79}{6}\right)^2 = \frac{6241}{36} \\ (-3 - \overline{x})^2 &= \left(-3 - \left(-\frac{23}{6}\right)\right)^2 = \left(\frac{5}{6}\right)^2 = \frac{25}{36} \\ (-11 - \overline{x})^2 &= \left(-11 - \left(-\frac{23}{6}\right)\right)^2 = \left(-\frac{43}{6}\right)^2 = \frac{1849}{36} \\ (-5 - \overline{x})^2 &= \left(-5 - \left(-\frac{23}{6}\right)\right)^2 = \left(-\frac{7}{6}\right)^2 = \frac{49}{36} \\ \end{align*} \] 3. **Sum the squared deviations**: \[ \sum (x_i - \overline{x})^2 = \frac{841 + 9025 + 6241 + 25 + 1849 + 49}{36} = \frac{18030}{36} = \frac{3005}{6} \] 4. **Calculate the sample variance (s²)**: \[ s^2 = \frac{\sum (x_i - \overline{x})^2}{n - 1} = \frac{\frac{3005}{6}}{5} = \frac{3005}{6 \times 5} = \frac{3005}{30} = \frac{601}{6} \] So, the sample variance of the data set is: \[\boxed{\dfrac{601}{6}}\] |
\frac{601}{6} |
| gpt-4o-2024-08-06 | To compute the sample variance, we follow these steps: 1. **Calculate the sample mean**: \[ \bar{x} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} = \frac{-23}{6} = -\frac{23}{6} \] 2. **Calculate each deviation from the mean, square it, and sum these squared deviations**: \[ \begin{align*} (1 - \bar{x})^2 &= \left(1 + \frac{23}{6}\right)^2 = \left(\frac{6}{6} + \frac{23}{6}\right)^2 = \left(\frac{29}{6}\right)^2 = \frac{841}{36}, \\ (12 - \bar{x})^2 &= \left(12 + \frac{23}{6}\right)^2 = \left(\frac{72}{6} + \frac{23}{6}\right)^2 = \left(\frac{95}{6}\right)^2 = \frac{9025}{36}, \\ (-17 - \bar{x})^2 &= \left(-17 + \frac{23}{6}\right)^2 = \left(-\frac{102}{6} + \frac{23}{6}\right)^2 = \left(-\frac{79}{6}\right)^2 = \frac{6241}{36}, \\ (-3 - \bar{x})^2 &= \left(-3 + \frac{23}{6}\right)^2 = \left(-\frac{18}{6} + \frac{23}{6}\right)^2 = \left(\frac{5}{6}\right)^2 = \frac{25}{36}, \\ (-11 - \bar{x})^2 &= \left(-11 + \frac{23}{6}\right)^2 = \left(-\frac{66}{6} + \frac{23}{6}\right)^2 = \left(-\frac{43}{6}\right)^2 = \frac{1849}{36}, \\ (-5 - \bar{x})^2 &= \left(-5 + \frac{23}{6}\right)^2 = \left(-\frac{30}{6} + \frac{23}{6}\right)^2 = \left(-\frac{7}{6}\right)^2 = \frac{49}{36}. \end{align*} \] 3. **Sum the squared deviations**: \[ \sum (x_i - \bar{x})^2 = \frac{841}{36} + \frac{9025}{36} + \frac{6241}{36} + \frac{25}{36} + \frac{1849}{36} + \frac{49}{36} = \frac{18230}{36} \] 4. **Calculate the sample variance**: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{\frac{18230}{36}}{5} = \frac{18230}{180} \] 5. **Simplify the fraction**: \[ \frac{18230}{180} = \frac{9115}{90} \] Thus, the sample variance is: \[ \boxed{\frac{9115}{90}} \] |
\frac{601}{6} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample variance of the list \({1, 12, -17, -3, -11, -5}\), we follow these steps: 1. **Calculate the mean (average) of the numbers:** \[ \text{Mean} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} \] \[ \text{Sum} = 1 + 12 - 17 - 3 - 11 - 5 = -23 \] \[ \text{Mean} = \frac{-23}{6} \approx -3.8333 \] 2. **Calculate the squared differences from the mean:** \[ (1 - (-3.8333))^2 = (1 + 3.8333)^2 = 4.8333^2 \] \[ (12 - (-3.8333))^2 = (12 + 3.8333)^2 = 15.8333^2 \] \[ (-17 - (-3.8333))^2 = (-17 + 3.8333)^2 = -13.1667^2 \] \[ (-3 - (-3.8333))^2 = (-3 + 3.8333)^2 = 0.8333^2 \] \[ (-11 - (-3.8333))^2 = (-11 + 3.8333)^2 = -7.1667^2 \] \[ (-5 - (-3.8333))^2 = (-5 + 3.8333)^2 = -1.1667^2 \] 3. **Compute the squared differences:** \[ (1 + 3.8333)^2 = 4.8333^2 = 23.3583 \] \[ (12 + 3.8333)^2 = 15.8333^2 = 250.6889 \] \[ (-17 + 3.8333)^2 = -13.1667^2 = 173.3389 \] \[ (-3 + 3.8333)^2 = 0.8333^2 = 0.6944 \] \[ (-11 + 3.8333)^2 = -7.1667^2 = 51.3389 \] \[ (-5 + 3.8333)^2 = -1.1667^2 = 1.3611 \] 4. **Sum the squared differences:** \[ \text{Sum of squared differences} = 23.3583 + 250.6889 + 173.3389 + 0.6944 + 51.3389 + 1.3611 \] \[ \text{Sum of squared differences} = 500.7795 \] 5. **Calculate the sample variance:** \[ \text{Sample variance} = \frac{500.7795}{6 - 1} = \frac{500.7795}{5} = 100.1559 \] Thus, the sample variance is: \[ \boxed{100.1559} \] |
\frac{601}{6} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Calculate the sample mean To find the sample variance, we first need to calculate the sample mean. The sample mean is the average of all the numbers in the dataset. The dataset is ${1, 12, -17, -3, -11, -5}$. We add up all the numbers and divide by the total count of numbers. ## Step 2: Sum up all the numbers in the dataset $1 + 12 - 17 - 3 - 11 - 5 = -23$ ## Step 3: Calculate the sample mean The total count of numbers is 6. The sample mean is $\frac{-23}{6}$. ## Step 4: Calculate the deviations from the mean We need to find the deviations of each number from the sample mean. The deviations are calculated as each number minus the sample mean. ## Step 5: Calculate the squared deviations The squared deviations are calculated as each deviation squared. ## Step 6: Calculate the sum of the squared deviations We add up all the squared deviations. ## Step 7: Calculate the sample variance The sample variance is the sum of the squared deviations divided by the total count of numbers minus 1. ## Step 8: Perform the calculations for steps 4-7 The sample mean is $\frac{-23}{6}$. The deviations are: - $1 - \frac{-23}{6} = 1 + \frac{23}{6} = \frac{29}{6}$ - $12 - \frac{-23}{6} = 12 + \frac{23}{6} = \frac{95}{6}$ - $-17 - \frac{-23}{6} = -17 + \frac{23}{6} = \frac{-79}{6}$ - $-3 - \frac{-23}{6} = -3 + \frac{23}{6} = \frac{11}{6}$ - $-11 - \frac{-23}{6} = -11 + \frac{23}{6} = \frac{-43}{6}$ - $-5 - \frac{-23}{6} = -5 + \frac{23}{6} = \frac{13}{6}$ The squared deviations are: - $(\frac{29}{6})^2 = \frac{841}{36}$ - $(\frac{95}{6})^2 = \frac{9025}{36}$ - $(\frac{-79}{6})^2 = \frac{6241}{36}$ - $(\frac{11}{6})^2 = \frac{121}{36}$ - $(\frac{-43}{6})^2 = \frac{1849}{36}$ - $(\frac{13}{6})^2 = \frac{169}{36}$ The sum of the squared deviations is: $\frac{841}{36} + \frac{9025}{36} + \frac{6241}{36} + \frac{121}{36} + \frac{1849}{36} + \frac{169}{36} = \frac{18246}{36}$ ## Step 9: Calculate the sample variance The sample variance is $\frac{18246}{36} \div (6 - 1) = \frac{18246}{36} \div 5 = \frac{18246}{180}$. ## Step 10: Simplify the sample variance $\frac{18246}{180} = 101.37$ The final answer is: $\boxed{101.37}$ |
\frac{601}{6} |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Calculate the sample mean To find the sample variance, we first need to calculate the sample mean. The sample mean is calculated by summing all the values and dividing by the number of values. The values given are ${1, 12, -17, -3, -11, -5}$. The sum of these values is $1 + 12 - 17 - 3 - 11 - 5 = -23$. There are $6$ values in total. Therefore, the sample mean is $-23 / 6 = -3.8333$. ## Step 2: Subtract the sample mean from each value Next, we subtract the sample mean from each value to find the deviations from the mean. The deviations are $(1 - (-3.8333))$, $(12 - (-3.8333))$, $(-17 - (-3.8333))$, $(-3 - (-3.8333))$, $(-11 - (-3.8333))$, and $(-5 - (-3.8333))$. This simplifies to $4.8333$, $15.8333$, $-13.1667$, $0.8333$, $-7.1667$, and $-1.1667$. ## Step 3: Square each deviation Now, we square each deviation. The squared deviations are $(4.8333)^2$, $(15.8333)^2$, $(-13.1667)^2$, $(0.8333)^2$, $(-7.1667)^2$, and $(-1.1667)^2$. This gives us $23.3056$, $250.3056$, $173.6111$, $0.6944$, $51.9444$, and $1.3611$. ## Step 4: Calculate the sum of the squared deviations We then calculate the sum of these squared deviations. The sum is $23.3056 + 250.3056 + 173.6111 + 0.6944 + 51.9444 + 1.3611 = 501.2222$. ## Step 5: Calculate the sample variance Finally, to find the sample variance, we divide the sum of the squared deviations by the number of values minus one (which is $6 - 1 = 5$). Therefore, the sample variance is $501.2222 / 5 = 100.2444$. The final answer is: $\boxed{100.2444}$ |
\frac{601}{6} |
| Qwen2-72B-Instruct | To compute the sample variance of a set of numbers, we first need to find the mean (average) of those numbers, then calculate the squared differences from the mean for each number, and finally take the average of those squared differences. Given the set of numbers: ${1, 12, -17, -3, -11, -5}$ First, let's find the mean: \[ \text{Mean} = \frac{\sum x_i}{n} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} = \frac{-23}{6} = -3.8333... \] Next, we calculate the squared differences from the mean for each number: \[ (1 - (-3.8333))^2 = 20.9722 \\ (12 - (-3.8333))^2 = 274.9722 \\ (-17 - (-3.8333))^2 = 176.1111 \\ (-3 - (-3.8333))^2 = 0.6944 \\ (-11 - (-3.8333))^2 = 51.1111 \\ (-5 - (-3.8333))^2 = 1.4815 \\ \] Then, we sum these squared differences and divide by \(n-1\) (since this is the sample variance formula): \[ \text{Sample Variance} = \frac{\sum(x_i - \bar{x})^2}{n-1} = \frac{20.9722 + 274.9722 + 176.1111 + 0.6944 + 51.1111 + 1.4815}{6-1} \\ = \frac{525.3425}{5} = 105.0685 \] Therefore, the sample variance of the given set of numbers is approximately \(105.0685\). \[ \boxed{105.0685} \] |
\frac{601}{6} |
| baichuan4 | To compute the sample variance of a set of numbers, we first need to find the mean (average) of the numbers. Then, for each number in the set, we subtract the mean from that number, square the result, and sum all these squared differences. Finally, we divide this sum by the number of elements in the set minus one (since we are dealing with sample variance, not population variance). Given the set of numbers: ${1, 12, -17, -3, -11, -5}$, let's calculate the sample variance step by step. 1. Calculate the mean: Mean = (1 + 12 - 17 - 3 - 11 - 5) / 6 = (-13) / 6 = -2.1667 (approximately) 2. For each number, subtract the mean and square the result: (1 - (-2.1667))^2 = (3.1667)^2 = 10.0001 (approximately) (12 - (-2.1667))^2 = (14.1667)^2 = 200.9996 (approximately) (-17 - (-2.1667))^2 = (-14.8333)^2 = 219.9992 (approximately) (-3 - (-2.1667))^2 = (-0.8333)^2 = 0.6944 (approximately) (-11 - (-2.1667))^2 = (-8.8333)^2 = 77.9992 (approximately) (-5 - (-2.1667))^2 = (-2.8333)^2 = 8.0001 (approximately) 3. Sum all the squared differences: 10.0001 + 200.9996 + 219.9992 + 0.6944 + 77.9992 + 8.0001 = 517.6926 (approximately) 4. Divide the sum by the number of elements minus one (since it's sample variance): Sample Variance = 517.6926 / (6 - 1) = 517.6926 / 5 = 103.5385approximately)approximately) Therefore, the sample variance of the given set of numbers is approximately 103.5385. We can put this answer in a box as requested: $ \boxed{103.5385} $ |
\frac{601}{6} |
| doubao | First find the mean: $\overline{x} = (1 + 12 + (-17) + (-3) + (-11) + (-5)) \div 6$ $= (13 - 17 - 3 - 11 - 5) \div 6$ $= (-4 - 3 - 11 - 5) \div 6$ $= (-7 - 11 - 5) \div 6$ $= (-18 - 5) \div 6$ $= -23 \div 6 = -\frac{23}{6}$. Then calculate the sum of squared deviations from the mean: \[ \begin{align*} &[(1 - (-\frac{23}{6}))^2 + (12 - (-\frac{23}{6}))^2 + ((-17) - (-\frac{23}{6}))^2 + ((-3) - (-\frac{23}{6}))^2 + ((-11) - (-\frac{23}{6}))^2 + ((-5) - (-\frac{23}{6}))^2]\\ =&[(\frac{29}{6})^2 + (\frac{95}{6})^2 + (-\frac{125}{6})^2 + (-\frac{5}{6})^2 + (-\frac{89}{6})^2 + (-\frac{53}{6})^2]\\ =&\frac{841}{36} + \frac{9025}{36} + \frac{15625}{36} + \frac{25}{36} + \frac{7921}{36} + \frac{2809}{36}\\ =&\frac{841 + 9025 + 15625 + 25 + 7921 + 2809}{36}\\ =&\frac{36246}{36} \end{align*} \] Finally, the sample variance is the sum of squared deviations from the mean divided by (n - 1), so the sample variance is: $\frac{36246}{36 \times (6 - 1)} = \frac{36246}{36 \times 5} = \frac{36246}{180} = \frac{6041}{30} = \boxed{\frac{6041}{30}}$. |
\frac{601}{6} |
| gpto1 | To find the sample variance of the given data set \{1, 12, -17, -3, -11, -5\}, we'll follow these steps: 1. **Calculate the sample mean (\(\overline{x}\))**: \[ \overline{x} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} = \frac{-23}{6} \] 2. **Find each squared deviation from the mean**: \[ \begin{align*} (1 - \overline{x})^2 &= \left(1 - \left(-\frac{23}{6}\right)\right)^2 = \left(\frac{29}{6}\right)^2 = \frac{841}{36} \\ (12 - \overline{x})^2 &= \left(12 - \left(-\frac{23}{6}\right)\right)^2 = \left(\frac{95}{6}\right)^2 = \frac{9025}{36} \\ (-17 - \overline{x})^2 &= \left(-17 - \left(-\frac{23}{6}\right)\right)^2 = \left(-\frac{79}{6}\right)^2 = \frac{6241}{36} \\ (-3 - \overline{x})^2 &= \left(-3 - \left(-\frac{23}{6}\right)\right)^2 = \left(\frac{5}{6}\right)^2 = \frac{25}{36} \\ (-11 - \overline{x})^2 &= \left(-11 - \left(-\frac{23}{6}\right)\right)^2 = \left(-\frac{43}{6}\right)^2 = \frac{1849}{36} \\ (-5 - \overline{x})^2 &= \left(-5 - \left(-\frac{23}{6}\right)\right)^2 = \left(-\frac{7}{6}\right)^2 = \frac{49}{36} \\ \end{align*} \] 3. **Sum the squared deviations**: \[ \sum (x_i - \overline{x})^2 = \frac{841 + 9025 + 6241 + 25 + 1849 + 49}{36} = \frac{18030}{36} = \frac{3005}{6} \] 4. **Calculate the sample variance (s²)**: \[ s^2 = \frac{\sum (x_i - \overline{x})^2}{n - 1} = \frac{\frac{3005}{6}}{5} = \frac{3005}{6 \times 5} = \frac{3005}{30} = \frac{601}{6} \] So, the sample variance of the data set is: \[\boxed{\dfrac{601}{6}}\] |
\frac{601}{6} |
| kimi | First, calculate the mean of the data set: $$\bar{x} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} = \frac{-23}{6}.$$ Next, calculate the squared differences from the mean and sum them up: \begin{align*} s^2 &= \frac{1}{6} \left[ (1 - \frac{-23}{6})^2 + (12 - \frac{-23}{6})^2 + (-17 - \frac{-23}{6})^2 + (-3 - \frac{-23}{6})^2 \right. \\ &\left. \quad + (-11 - \frac{-23}{6})^2 + (-5 - \frac{-23}{6})^2 \right] \\ &= \frac{1}{6} \left[ \left(\frac{29}{6}\right)^2 + \left(\frac{59}{6}\right)^2 + \left(\frac{-71}{6}\right)^2 + \left(\frac{-11}{6}\right)^2 \right. \\ &\left. \quad + \left(\frac{-49}{6}\right)^2 + \left(\frac{-43}{6}\right)^2 \right] \\ &= \frac{1}{6} \left[ \frac{841}{36} + \frac{3481}{36} + \frac{5041}{36} + \frac{121}{36} + \frac{2401}{36} + \frac{1849}{36} \right] \\ &= \frac{1}{6} \left[ \frac{13653}{36} \right] \\ &= \frac{13653}{216}. \end{align*} Thus, the sample variance is $\boxed{\frac{13653}{216}}$. |
\frac{601}{6} |
| minimax | To compute the sample variance of the given data set ${1, 12, -17, -3, -11, -5}$, we will follow these steps: 1. Find the mean (average) of the data set. 2. Subtract the mean from each data point to find the deviations. 3. Square each deviation. 4. Sum all the squared deviations. 5. Divide the sum by $n - 1$, where $n$ is the number of data points. Let's go through these steps: 1. Calculate the mean: $$ \text{mean} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} = \frac{-13}{6} = -2.1667 $$ 2. Calculate the deviations from the mean: $$ 1 - (-2.1667) = 3.1667 $$ $$ 12 - (-2.1667) = 14.1667 $$ $$ -17 - (-2.1667) = -14.8333 $$ $$ -3 - (-2. |
\frac{601}{6} |
| wenxinyiyan | 计算过程如下： 首先，我们需要计算样本的平均值。样本平均值的计算公式是： 平均值 = (样本1 + 样本2 + ... + 样本n) / n 对于给定的样本 {1, 12, -17, -3, -11, -5}，我们有 n = 6。 平均值 = (1 + 12 - 17 - 3 - 11 - 5) / 6 = (-13) / 6 = -13/6 接下来，我们计算每个样本值与平均值的差的平方，然后求这些平方差的平均值。样本方差的计算公式是： 样本方差 = Σ(样本值 - 平均值)^2 / (n - 1) 其中，Σ 表示求和，n 是样本数量。 样本方差 = [(1 + 13/6)^2 + (12 + 13/6)^2 + (-17 + 13/6)^2 + (-3 + 13/6)^2 + (-11 + 13/6)^2 + (-5 + 13/6)^2] / (6 - 1) = [(19/6)^2 + (85/6)^2 + (-109/6)^2 + (-21/6)^2 + (-79/6)^2 + (-37/6)^2] / 5 = [(361/36) + (7225/36) + (11881/36) + (441/36) + (6241/36) + (1369/36)] / 5 = (27548/36) / 5 = (765.2222...) / 5 = 153.0444... 所以，样本方差约为 153.04。 最终答案： 样本方差 = 153.04（四舍五入到小数点后两位） |
\frac{601}{6} |
| zhipu | To find the sample variance of the set ${1, 12, -17, -3, -11, -5}$, we follow these steps: 1. Calculate the sample mean (average) of the data set. 2. Subtract the sample mean from each data point to find the deviations. 3. Square each deviation. 4. Sum the squared deviations. 5. Divide the sum of the squared deviations by the degrees of freedom (which is the number of data points minus 1). **Step 1: Calculate the sample mean:** The sample mean (\(\bar{x}\)) is given by: \[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \] where \(n\) is the number of data points and \(x_i\) are the data points. For our data set: \[ \bar{x} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} = \frac{-23}{6} = -\frac{23}{6} \] **Step 2: Calculate the deviations from the mean:** \[ 1 - \left(-\frac{23}{6}\right), \quad 12 - \left(-\frac{23}{6}\right), \quad -17 - \left(-\frac{23}{6}\right), \quad -3 - \left(-\frac{23}{6}\right), \quad -11 - \left(-\frac{23}{6}\right), \quad -5 - \left(-\frac{23}{6}\right) \] Simplify each deviation: \[ 1 + \frac{23}{6} = \frac{6}{6} + \frac{23}{6} = \frac{29}{6} \] \[ 12 + \frac{23}{6} = \frac{72}{6} + \frac{23}{6} = \frac{95}{6} \] \[ -17 + \frac{23}{6} = -\frac{102}{6} + \frac{23}{6} = -\frac{79}{6} \] \[ -3 + \frac{23}{6} = -\frac{18}{6} + \frac{23}{6} = \frac{5}{6} \] \[ -11 + \frac{23}{6} = -\frac{66}{6} + \frac{23}{6} = -\frac{43}{6} \] \[ -5 + \frac{23}{6} = -\frac{30}{6} + \frac{23}{6} = -\frac{7}{6} \] **Step 3: Square each deviation:** \[ \left(\frac{29}{6}\right)^2 = \frac{841}{36} \] \[ \left(\frac{95}{6}\right)^2 = \frac{9025}{36} \] \[ \left(-\frac{79}{6}\right)^2 = \frac{6241}{36} \] \[ \left(\frac{5}{6}\right)^2 = \frac{25}{36} \] \[ \left(-\frac{43}{6}\right)^2 = \frac{1849}{36} \] \[ \left(-\frac{7}{6}\right)^2 = \frac{49}{36} \] **Step 4: Sum the squared deviations:** \[ \frac{841}{36} + \frac{9025}{36} + \frac{6241}{36} + \frac{25}{36} + \frac{1849}{36} + \frac{49}{36} = \frac{841 + 9025 + 6241 + 25 + 1849 + 49}{36} = \frac{18030}{36} = 500.8333 \] **Step 5: Divide by the degrees of freedom (n-1):** The degrees of freedom is \(6 - 1 = 5\). \[ \text{Sample Variance} = \frac{\sum (\text{Squared Deviations})}{n-1} = \frac{500.8333}{5} = 100.1667 \] So, the sample variance of the set ${1, 12, -17, -3, -11, -5}$ is: \[ \boxed{100.1667} \] |
\frac{601}{6} |
| deepseek-v3 | To compute the **sample variance** of the dataset \(\{1, 12, -17, -3, -11, -5\}\), follow these steps: 1. **Calculate the Mean (\(\bar{x}\)):** \[ \bar{x} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} = \frac{-23}{6} \approx -3.8333 \] 2. **Find the Squared Differences from the Mean:** \[ \begin{align*} (1 - (-3.8333))^2 &= (4.8333)^2 \approx 23.361 \\ (12 - (-3.8333))^2 &= (15.8333)^2 \approx 250.694 \\ (-17 - (-3.8333))^2 &= (-13.1667)^2 \approx 173.361 \\ (-3 - (-3.8333))^2 &= (0.8333)^2 \approx 0.694 \\ (-11 - (-3.8333))^2 &= (-7.1667)^2 \approx 51.361 \\ (-5 - (-3.8333))^2 &= (-1.1667)^2 \approx 1.361 \\ \end{align*} \] 3. **Sum of Squared Differences:** \[ 23.361 + 250.694 + 173.361 + 0.694 + 51.361 + 1.361 \approx 500.832 \] 4. **Compute the Sample Variance (\(s^2\)):** \[ s^2 = \frac{\text{Sum of Squared Differences}}{n - 1} = \frac{500.832}{5} \approx 100.1664 \] Rounding to two decimal places: \[ s^2 \approx 100.17 \] Therefore, the sample variance is: \[ \boxed{100.17} \] |
\frac{601}{6} |